Scale and tempo of brain expansion in the course of human evolution implies that this process was driven by a positive feedback. The ‘cultural drive’ hypothesis suggests a possible mechanism for the runaway brain-culture coevolution wherein high-fidelity social learning results in accumulation of cultural traditions which, in turn, promote selection for still more efficient social learning. Here we explore this evolutionary mechanism by means of computer modeling. Simulations confirm its plausibility in a social species in a socio-ecological situation that makes the sporadic invention of new beneficial and cognitively demanding behaviours possible. The chances for the runaway brain-culture coevolution increase when some of the culturally transmitted behaviours are individually beneficial while the others are group-beneficial. In this case, ‘cultural drive’ is possible under varying levels of between-group competition and migration. Modeling implies that brain expansion can receive additional boost if the evolving mechanisms of social learning are costly in terms of brain expansion (e.g., rely on complex neuronal curcuits) and tolerant to the complexity of information transferred, that is, make it possible to transfer complex skills and concepts easily. Human language presumably fits this description. Modeling also confirms that the runaway brain-culture coevolution can be accelerated by additional positive feedback loops via population growth and lifespan extension, and that between-group competition and cultural group selection can facilitate the propagation of group-beneficial behaviours and remove maladaptive cultural traditions from the population’s culture, which individual selection is unable to do.